42
THEORY
OF
THERMAL
EQUILIBRIUM
§7.
Ideal
gases.
Absolute
temperature
The theory
we
developed
contains
as a
special
case
Maxwell's distribu-
tion
of states
for ideal
gases.
I.e.,
if in
§3
we
understand
by
the
system
S
one
gas
molecule
and
by
S
the
totality
of
all the others, then the
expres-
sion for the
probability
that the values
of
the variables
P1...Pn
of
S
lie
in
a
region
g
that is infinitesimally small with respect
to
all variables
will
be
d¥
=
Ae~2hE dpv..dq
g
1
*
One
can
also
immediately
realize
from
the
expression
for the
quantity
h
found
in
§4
that,
up
to
the
infinitesimally small,
the
quantity
h
will
be
the
same
for
a
gas
molecule of
another
type occuring
in
the
system,
since the
systems S determining h
are
identical for the
two
molecules
up
to
the
infinitesimally small. This establishes the
generalized Maxwellian
distribution
of states
for ideal
gases.
-
Further,
it follows
immediately
that the
mean
kinetic
energy
of motion
of
the
center of gravity of
a
gas
molecule
occurring
in
a
system
S
has the
value
3/4
h
because it
corresponds
to
three
momentoids.
The
kinetic
theory
of
gases
teaches
us
that this
quantity
is proportional
to
the
gas pressure
at
constant
volume.
If,
by
definition,
this is taken
to be proportional to
the
absolute
temperature,
one
obtains
a
relationship of
the
form
1
-
71
_
i
ftj(E)
TK
~
K'
1
~
2 -
J
0/'(E)
[25]
where
K
denotes
a
universal
constant, and
w
the function introduced
in
§3.
§8. The
second
law of
the
theory of
heat
as a
consequence
of the
mechanical
theory
We
consider
a
given
physical
system S
as a
mechanical
system
with
coordinates
p1...pn. As
state
variables
of
the
system we
further introduce
the
quantities